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I had a thought to estimate how many times a virus is transmitted, on average, before infecting a given individual.

For instance, if Sample Virus has a re-transmission rate of 2, and a total infected population of 15, we can guess the virus has transmitted 3 times. (1+2+4+8)

Given a re-transmission rate of 1.2, and a global infected population of 54 million, how many people can we estimate the COVID-19 virus has "gone through" before someone who has not been previously infected now contracts the virus?

This is not a homework question but I am having a lot of trouble finding a formula for this. I used Excel auto-complete to repeatedly multiply the current population by 1.2 and found an answer of about 85 transmissions. Is this the most accurate method here? How can one find a formula for this answer?

Scott Sinischo
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  • It is really an interesting question. I am looking forward for the other approaches as well. – blackmirror7 Nov 15 '20 at 21:07
  • "how many people can we estimate the COVID-19 virus has gone through" : I'm not sure what you are asking. Your math suggests that we are to assume the following: Start with a patient 0 (denote as Level 0). Label each person infected by him as a Level 1 patient. Label each person infected by a Level $n$ patient as a Level $(n+1)$ patient. Assume that each Level $n$ patient infects $\approx$ 1.2 people, and then stops infecting people. Then how many Levels of patients are represented by 54 million people. ...see next comment – user2661923 Nov 16 '20 at 12:49
  • Note that $1 + x + x^2 + \cdots + x^k ~=~ \frac{x^{(k+1)} - 1}{x - 1}.$ Therefore, you are looking for $(k)$ such that $\frac{(1.2)^{(k+1)} - 1}{1.2 - 1} \approx 54000000 \implies (1.2)^{(k+1)} \approx (0.2) \times 54000000.$ Taking the Log of both sides gives $(k+1) \approx \frac{[\log ~0.2] + [\log ~54000000]}{\log ~1.2} \approx 88.8 \implies ~k \approx 88.$ If you are asking a different question and want to flag me, address a comment to @user2661923. – user2661923 Nov 16 '20 at 12:57
  • @user2661923 Hello, thank you for taking a look. This was the question I was asking indeed and I think this is the most straightforward way to analyze it. I recognize this is a series - what is the name for this particular one so I can brush up on it? I am also curious if someone can expand on this, considering the available non-infected population decreases over time which may increase the ultimate result. Does anyone know if similar calculations exist in the medical field for tracking things like mutation chance, etc.? – Scott Sinischo Nov 16 '20 at 20:03
  • See geometric series. As for your other questions, since I am totally ignorant here, all I can do is refer you to Epidemiology. – user2661923 Nov 16 '20 at 21:42

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